How can I edit the city / hamlet that some newly added streets I have mapped are showing under? I need to remove the hamlet because it is incorrect and should be showing under a city instead.

Scenario: one street I added is showing as Evans Woods Drive, Orchard Park, Mecklenburg County, NC
This is incorrect. This address is deeded as Charlotte, NC and is not listed under the Orchard Park hamlet.

With that being said, how do I correct the address for the street I have added to show up as Charlotte.

That coincidence makes me wonder if "place=hamlet" has been mapped differently in Charlotte than other cities in the USA. Certainly from a European point of view it seems odd to have a hamlet inside a city, but I know there are certain practices in OSM to reflect specifically American settlement types or administrative areas.

It is quite close to your address so I would guess Nominatim, the OSM search engine, is giving it priority over the city based on proximity. Note that Nominatim has to try to give reasonable answers for all countries in the world, with all their different address systems. I wouldn't necessarily expect it to consistently match what is shown on title deeds even if all underlying data is perfectly correct. But in this case it does seem worth looking further into the choice of place=hamlet, maybe asking in a USA-specific forum or mailing list.

By the way I note that you have changed the name of https://www.openstreetmap.org/way/665624338 to "Evans Woods Drive, Charlotte, NC". That is definitely NOT the solution - the name tag should not include things that are not part of the name, such a non-standard approach won't help you in the long run.

I also notice that this street is outside the boundary of Charlotte (https://www.openstreetmap.org/relation/177415 ) - you can see that this boundary is a bit to the south roughly following a railway line. That may well be contributing to Nominatim giving priority to the nearby hamlet rather than the city. You say this street is in Charlotte, so maybe it is the city boundary that is the issue?

I assume you are talking about the search result that you get displayed when searching for the street on www.openstreetmap.org.

Don't worry too much about what is shown there. The search engine (called Nominatim) does some guessing when it comes to how objects are hierarchically related with administrative bodies. It uses information from place nodes (place=hamlet, place=suburb) and other data but frankly speaking does a lousy job in guessing accurately in many cases. To really indicate where an object is located inside an administrative boundary it needs to be mapped as such. But even then Nominatim is still making errors. As alan_gr pointed out, don't try to improve the search result display by adding information to the name that is not the name.